Long story short: our goal is to describe tail decay and moderate deviation estimates for the distribution of the height function of the stochastic six-vertex model, with narrow wedge initial condition.

Now the long version. This whole tale of the tail started in Ann Arbor, many many years ago. Explaining the embarrassment of our failure to find such estimates for so many years is the main reason for our talk. As we had to learn for our task, such tail estimates are intimately connected with conditional thinning ensembles from random matrix theory, and nonlocal versions of the so-called Painlevé equations. The unraveling of such connections was a parallel task that we had to overcome, together with many other colleagues, and led to interesting phenomena that we hadn't anticipated with our original question. During our talk, we plan to go through such developments as well.

The talk is mostly based on joint works with Promit Ghosal (University of Chicago), but partially also in works in preparation with Tom Claeys (UC Louvain), Leslie Molag (Universidad Carlos III) and Lun Zhang (Fudan University).